Treating neural network inputs and outputs as random variables, we
characterize the structure of neural networks that can be used to model data
that are invariant or equivariant under the action of a compact group. Much
recent research has been devoted to encoding invariance under symmetry
transformations into neural network architectures, in an effort to improve the
performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised
settings. By considering group invariance from the perspective of probabilistic
symmetry, we establish a link between functional and probabilistic symmetry,
and obtain generative functional representations of probability distributions
that are invariant or equivariant under the action of a compact group. Our
representations completely characterize the structure of neural networks that
can be used to model such distributions and yield a general program for
constructing invariant stochastic or deterministic neural networks. We
demonstrate that examples from the recent literature are special cases, and
develop the details of the general program for exchangeable sequences and
arrays.

本研究从概率对称性的角度考虑群不变性，建立功能性和概率对称性之间的联系，并得到了不变或等变于紧致群作用下的概率分布的生成功能表示。此表示完全表征了神经网络的结构，可用于模拟此类分布并提供了一般性的计算程序。

概率对称和不变神经网络

Probabilistic symmetries and invariant neural networks

Convolutional neural networks have been extremely successful in the image
recognition domain because they ensure equivariance to translations. There have
been many recent attempts to generalize this framework to other domains,
including graphs and data lying on manifolds. In this paper we give a rigorous,
theoretical treatment of convolution and equivariance in neural networks with
respect to not just translations, but the action of any compact group. Our main
result is to prove that (given some natural constraints) convolutional
structure is not just a sufficient, but also a necessary condition for
equivariance to the action of a compact group. Our exposition makes use of
concepts from representation theory and noncommutative harmonic analysis and
derives new generalized convolution formulae.

本文首先说明深度学习中卷积神经网络在图像识别方面的成功，然后介绍了将此框架推广到其他领域的尝试，并给出了使用表示论和非交换调和分析概念的卷积和等变性的理论证明，证明了卷积结构不仅是充分而且也是必要的对于紧致群作用的等变性的条件，并推导了新的广义卷积公式。